Aristotle - The Organon - index for ANALYTICA PRIORIA Book 1 Part 45

Reduction of syllogisms between figures

Paragraph 1 Whatever problems are proved in more than one figure, if they have been established in one figure by syllogism, can be reduced to another figure, e.g. a negative syllogism in the first figure can be reduced to the second, and a syllogism in the middle figure to the first, not all however but some only.
Paragraph 2 The universal syllogisms in the second figure can be reduced to the first, but only one of the two particular syllogisms.
Paragraph 3 But if the syllogism is particular, whenever the negative statement concerns the major extreme, reduction to the first figure will be possible, e.g. if A belongs to no B and to some C:
Paragraph 4 Again syllogisms in the third figure cannot all be resolved into the first, though all syllogisms in the first figure can be resolved into the third.
Paragraph 5 Of the syllogisms in the last figure one only cannot be resolved into the first, viz. when the negative statement is not universal:
Paragraph 6 It is clear that in order to resolve the figures into one another the premiss which concerns the minor extreme must be converted in both the figures:
Paragraph 7 One of the syllogisms in the middle figure can, the other cannot, be resolved into the third figure.
Paragraph 8 Syllogisms in the third figure can be resolved into the middle figure, whenever the negative statement is universal, e.g. if A belongs to no C, and B to some or all C.
Paragraph 9 It is clear then that the same syllogisms cannot be resolved in these figures which could not be resolved into the first figure, and that when syllogisms are reduced to the first figure these alone are confirmed by reduction to what is impossible.
Paragraph 10 It is clear from what we have said how we ought to reduce syllogisms, and that the figures may be resolved into one another.

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